answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of ฯ is not used.
step-by-step explanation:
radius is diameter/2 = 4.
area of circle is ฯ r^2.
volume of cylinder is height ร area of base = hฯr^2 = 3(4^2) ฯ = 48ฯ cubic inches.
not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).
volume of cone is integral 0 to h area(y) dy. area(y) is ฯ r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.
r(y) = r - r/h y
area(y) = ฯ (r - r/h y)^2 = ฯ (r^2 - 2r^2y/h + r^2y^2/h^2) = ฯ r^2(1-2y/h+y^2/h^2)
antiderivative area(y) = ฯ r^2 (y-y^2/h+y^3/(3h^2))
integral 0 to h = ฯ r^2(h-h^2/h+h^3/(3h^2) - (0)) = ฯ r^2(h - h + h/3) = ฯ r^2(h/3) = ฯ/3 r^2 h
volume of this cone is ฯ/3(4^2)18 cubic inches.
= 6 16 ฯ = 96ฯ cubic inches.
cone has twice the volume, 96/48.
cone height is 6 times the cylinder.
ratio of volume formula: ฯ r^2 cancels.
cone over cylinder ratio is 6h/3 to h, 2 to 1.
